Optimal. Leaf size=71 \[ \frac {2 \left (\frac {1}{x}+1\right )^{3/2} \sqrt {1-x} x^2}{5 \sqrt {1-\frac {1}{x}}}-\frac {4 \left (\frac {1}{x}+1\right )^{3/2} \sqrt {1-x} x}{15 \sqrt {1-\frac {1}{x}}} \]
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Rubi [A] time = 0.10, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {6176, 6181, 45, 37} \[ \frac {2 \left (\frac {1}{x}+1\right )^{3/2} \sqrt {1-x} x^2}{5 \sqrt {1-\frac {1}{x}}}-\frac {4 \left (\frac {1}{x}+1\right )^{3/2} \sqrt {1-x} x}{15 \sqrt {1-\frac {1}{x}}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 6176
Rule 6181
Rubi steps
\begin {align*} \int e^{\coth ^{-1}(x)} \sqrt {1-x} x \, dx &=\frac {\sqrt {1-x} \int e^{\coth ^{-1}(x)} \sqrt {1-\frac {1}{x}} x^{3/2} \, dx}{\sqrt {1-\frac {1}{x}} \sqrt {x}}\\ &=-\frac {\left (\sqrt {1-x} \sqrt {\frac {1}{x}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x}}{x^{7/2}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{x}}}\\ &=\frac {2 \left (1+\frac {1}{x}\right )^{3/2} \sqrt {1-x} x^2}{5 \sqrt {1-\frac {1}{x}}}+\frac {\left (2 \sqrt {1-x} \sqrt {\frac {1}{x}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x}}{x^{5/2}} \, dx,x,\frac {1}{x}\right )}{5 \sqrt {1-\frac {1}{x}}}\\ &=-\frac {4 \left (1+\frac {1}{x}\right )^{3/2} \sqrt {1-x} x}{15 \sqrt {1-\frac {1}{x}}}+\frac {2 \left (1+\frac {1}{x}\right )^{3/2} \sqrt {1-x} x^2}{5 \sqrt {1-\frac {1}{x}}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 41, normalized size = 0.58 \[ \frac {2 \sqrt {\frac {1}{x}+1} \sqrt {1-x} (x+1) (3 x-2)}{15 \sqrt {\frac {x-1}{x}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 40, normalized size = 0.56 \[ \frac {2 \, {\left (3 \, x^{3} + 4 \, x^{2} - x - 2\right )} \sqrt {-x + 1} \sqrt {\frac {x - 1}{x + 1}}}{15 \, {\left (x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.18, size = 44, normalized size = 0.62 \[ \frac {1}{15} \, {\left (-4 i \, \sqrt {2} - \frac {2 \, {\left (3 \, {\left (x + 1\right )}^{2} \sqrt {-x - 1} + 5 \, {\left (-x - 1\right )}^{\frac {3}{2}}\right )}}{\mathrm {sgn}\left (-x - 1\right )}\right )} \mathrm {sgn}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 29, normalized size = 0.41 \[ \frac {2 \left (1+x \right ) \left (3 x -2\right ) \sqrt {1-x}}{15 \sqrt {\frac {-1+x}{1+x}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.28, size = 17, normalized size = 0.24 \[ \frac {1}{15} \, {\left (6 i \, x^{2} + 2 i \, x - 4 i\right )} \sqrt {x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.27, size = 30, normalized size = 0.42 \[ -\frac {2\,\left (3\,x-2\right )\,\sqrt {\frac {x-1}{x+1}}\,{\left (x+1\right )}^2}{15\,\sqrt {1-x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 16.15, size = 46, normalized size = 0.65 \[ - \frac {14 i x}{15 \sqrt {\frac {1}{x + 1}}} - \frac {2 i \left (1 - x\right )^{2}}{5 \sqrt {\frac {1}{x + 1}}} + \frac {2 i}{3 \sqrt {\frac {1}{x + 1}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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